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't'ha. k$ lda soqn r HOAXC CEtiNG
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wfu cauv rY Ii$I 'cac BA^N 7'N YEU TAAN
c{c ttsng th&c & (1), (2), (3) vn (4) x&v ra khi
vi chl khi tam gidc ABC li tam giAc illu vA il/
!6. tem cha tam gi6c i16.
p0 th{y rn6i qu*+r tr$ gi&a c6c bdt d&ng thtc
.tr0n, ta cla bi€t th6 nio lri $i6 trf trung binh
cria c6c s6 duong. Igudi ta ggi gi{ tri trung
blnh bqc kr(k lil mQt sd thgc) cia eac s6 duong
f,I,:1ra,..., t'o II su
M x' (rr, xz,'.., ,,)-?GT;E;J;TI6'-
\rdi"dinh nghia d6 ta'th{y ngay bf,t d&ng thric
(l) c6 tlrB viCt dtr0i dqng
,lrr (1i., lll, Ro) > 2 Mt k)r, db, dc)' (1)
Cdu c6c uf;t d[tS tbtrc.(D,{3), (+) cnng dE ddng
itrra ilufc vE daug i ..
.Mt(Ra, Rr, ic) V z Mx(a., du, dd. (rr
tro-ng iI6 "k lBn luqt nlrQLn.cfe gid tri C,I12 r'l
l
Nhu v[Y, mQt gi& tliiEt rluqc d{t ra '
b?lt d&ng ii*" til ,i afins ubi mqi sd fft+'
''6t:
PIIAT TRIhN THfiM gfiT PAgC TE{TTC EC-B*-S#
NcirvfiN cONc QUt
.:
tf}AT d&ug ttrue'lic-d6-ss dn duqc gi0i thrQu
lp lrong !r5o 'lo6"n hr.rc v& tu&i tr6 s6 I * 10
n{m 1Q74. Cd ihF torn t6.t'nQi dunu bdi bio
d0.uhu sau,- NLrn tgfs nfre tohn iqc Hiig-ga-r'i
i)- 0c-,10-scr, ilor:g khr nghi0n irlu c6c itnh chf,t
cria'c6.c hinh phfrrg r1e kE xu{t m$t bfrt dfthg
- ' thr?c (sau niry Inaug te rr 0$g) nhu sau:
!- ,1, *f?l ! Rc> 2 ({3 + 6t6 I dc) (tl
trguil d; ni, Et, /ic l1i khoang ci,tb t* mEi alEm
t M n'im lrong rnXt ph&ng cda m$t lam gi6c ;{8r'
'?6n cAe rlinh cira tarrL gi6e: du, rlb, r1,' l[ khoing
' eich t& dibrn M ddn crrc canh cirn trm gi6c dd"
il ,rAm sau ugu{ri ta d* chftng miuh dtrqc }rit'
dfi,rg lhitc nr\y. (Trong b[i llAo c6 cho m$t chttng
. m!nh,.srv c{p cria b[t diing tit.uc). rt6 sau \r'gu]i
ta uott chitiiE mlnh duqc nbtng'b{'t dfirrg th&r:
' co litn cluar €dr b{t dlne thuc (1) ; fi6 la
t.
Tr&i qua nhitng sg tlur iOi nghi&n crlu, nhd
to$n hqc Ao Florlan nam 1958 di chung rninh
r&ng b{t tl&ng thuc (5) d6ng vA chl r}rtng veli
nhftng gid tri c&n /c € l. * t, t.l, tir dbng thoi e &ng
chung minh duEc ring vr'ri nhfi'ng s6 tirqrc
/c khdng.thu$" [ * 1, 1] ttli tn lu6n lu0n e6 bdi
<I&ag ihuc: .
tr,l
inr
I(k{R*, R6, ,tc) > Vt .4{p {d", dr, d6). (6)
Ti6p s*tl ddr bf,t rt*,ng thttc (t) d& tlugc m&
r$rrg vir* cta gi{c lili vir trcng khOng gian, niir
u ilo ta SA tilrs tlirqc nhirng k6t qu& ttrrJr vl'
Ntli tiip uh&ng ;d1 ng,hi !r0n. sart dtAy.si n0rr
l0n m&L ph*otg hu<llg ptr6.i. tribd b.h:rt cri"a
nhtrng i;!ilt rtirng tintc tlfl I irrr dutyc.
Gi:i sfr ,i5 c ho tarn gi6c llSC vil mtlt diBrn -11
nLrn trong :nIt ph&ng,cria tanr grAc fio (hinh vF).
A
Thlrc hiQn pbdp nghich dEo eilc M phuong. tich
l, ta thu duqc tam giAc A'B'C'. irong d6 A', B',
C' theo thtt tU lI &nh erla l, B v{ C trorig
pb€pugbich rliro. f* MB .I[B':,VC ' itfC' ta suy
i" tt*iwt:'=:vi{;lMr3'. Ti I+ th*c nay ehitng t6
ri.rrg hni tarn gii.e M-BC v* MC't]' (r:6 ch-ung
gric & dinh M) I?t dbug dang vdi nhau. "l'f s6
crLa ptr6p d6ng d4ng bi6n tarn giric thir nh$t
thhntr lem gi{c thir hai lh: d,
h{B' ME -|YIB' 1 I
!,{(: jlfB. }fC NtB . MC ftl lt*
-N6u dd.t MA':R'a, M8':J?n', MC'*ft'e vA gqi
.d'n, cl'b, i'*.th*o thu tqr ld kiro$ng.cAch tu diEm rlf
ddncee cqirh cfia tam gi&r A' B' C' thi ti'sd dbng
ttrqng ndi ir€n sE lit d'ridu, t{rc li" ta cd
}l{t kh6c ta.ed MA.MA' :1 truy ftqR'2: i,
tir d6 co .Ri -'llR, hay :
'iro'ng t u
B{ty giir gi6 nit trong lam gi6c ABC ta di. tim
rq ituEc mOL quan hd n*ro dei bi$u thi bflng mQt
b{t d&ng t}rL'rc giita .Ra, Rl, fi., cls, d5, dg' gi& sri
lar a ) $ (9). Yi ,{BC ld m$t tafir gi6c b.4.t k]
n€n b{t d&ng thirc aIE iir;r ra ei.ng dlirng cho
tam gi{c A'B'C'. tsXrg cAch thay cac giA tri c&a
rri, rli, rig vn I'li, iU, n; & (i) vii (8) r'?re l;{t
d&ng thire (9) ta s0 thu dnc.rc mOi b{t,3&ag lhuc
moi. I;ib don gi&n c6c ptrep tinir ta nliin :et
ring hai vd ct v& p eiia b{t tlLug th{re (0) bao
gi$ ctng ding edp v$ cung bflc el6i ','6i c:;c dq-i
d{i .Ra, l?r, llrn dn, dh, d". Do d6 ta ed il,6 griir
u&c m5"u s6 ehung Rufillic & c&c c6ag thue (7)
vd (s). I{.dt qui Ii ia di ddn nguyen tSc sau
di_v:
Gii sft trong m$'r tamgi5.c tae6 m$t quan hQ
oio dd biEu thi bing rrrf t bdt di.tg thue (hay
d&ng thile) gita eae d0 dil i?u, Rb, Rs, d,, db,
an. ll6u ttrqc hi*n tr8n b*{ d&ng ihtc (ha;,' d[ng
ih*c) d6 phdp l:i6u cibi /:
fia * l?tRc, l?l -'ll"iii;, li. + 1i"I?5,
O (ir'+ daIia, db, ** dt!i:, lic *'-F ti,:/1r,
thi ia sE thu 6uqc mQt birt d&ng thirc (hay
d&ng thuc) bi$u thi m$l qr:rn hQ nrc'i cflng dirng
trong tam gi{c.
EB minhhqrnguy€n ilc trdn ta h1ythte hi€n
phep bidn rl&i 7 tr6n b{t d&ng thr'rc (t) Trle
tttE" t" tliu ituqc'l.rdt d&r:g thuc
Rnfie*ficil"*R".Rr, )2(dnfiu+db1i1*d,:11"). (10)
Tuong tg, thgc iridn phep bi€n dbi / "trdl
cdc bt{t' <I&ng thitc (3) va (a) ta lhn iuq't thu
tlunc :
rl^qin. + ln;ii; +"\1R,ru>
> i2d rt,R. -f- l,irnn + :,lttrR"),
2(Ul?rRc + 1/.RcEr + i/Hafib) <
< llduns -[ lfuiir.Rt -l 1/dc]?c' '
/li = IirX,/It*/?ri?c \
fiL = E, Pul/?"Brll" [ ,t,
/ii = ,liufiu lla,Rnllc t
;}
\
'j
,i,
ea
(11)
(12)
Cac b6i iI&ng thitc'(10), (11) va {l2i ehi 1A
nhirug tru&ng h'1p d[c bj6t cfia m$t b{..t {&ng
thtrc i&ug rluat suu clty ihu duqc trl U6t tling
thec (5) erl.a Florian bXng e{ch thqc hi*s ir6n
tI6 ph6p bidn ai5i f :
cii/du * l/'R'/?c
tu fid! di: ds/fi5ft6
" py 6i = dufiul,llo.RlB. )
'i,.'{ trs rtri - dr"tir/nsfir,R. | {7)
A*: A"Rr/fi"rtlJ?. I
Mg(fitRc, ReRar Ila.Rr) 2
2Mx(dEfir, du.l?x, dciir), - 1<k < 1"
.Itt 11.
.6 : 128.8
(13)
\-
ri\
Ddi soi rrh&ng s6 ihge k)1 hole Jc< *1,
thrrc hi€n ph6p bi6n iiisn d8i / rsri hiit d&r-rg
thri,: r6) r(t l'luriau ta cu
,r,ih(Hr,P". "&,,tig, ,Qaftl) )
Itl (14)
7{i.rwv(r{alia, drnl, dc[i"),
Ch&ng h6n vfi lt':2 vir k: -Z ta lhn l*qt cir
nin? +niai+r3ni>
), tiri?;nf, + at ni + {i3fi*2). (15)
{ (r /,ilrr-i -i'r r,ri;;?, + li,Jlry?). ( } ti)
B6t rlirg thirc (i-i) cung- t,Ltr *ae tr'-trong hr.r1i
dac bi*i e&a nd la cdre bat d8.ag ihfte (i5), (ltjJ
khdng th6 ,< tang cuorg > itoti duqrc nir* ; nr,ri
c*,ch lihir, d5i voi bIt d&ng ihitc (14], ti.'sd
,'l{r (liufi". 6"Jf", Iir.fia) I ,t/r tr{o}iu, r/ufil' dsl?e)
' lu'l-
crl thE tidu dir^n tdi 'l/i: i:ao nhi6u .ciing dtuqe,
nhung khdng hao'gi& l,iitg rir:g'* gilr tr! d6.
Cde hqn th*.:.: mdu t 'I'r&n d:iy cilr'rng i* dii uSl
lAir m$i. pirrrong huSng ph:it siln th6nr bdi
ditng thu'c flc-$6-scr, llong r&ng chung ta ti6p
s*e nhan suy nghi, ndu lEn duqe nhftng ph'-rung
hu6'r:g lih6e cta vi6c *.qhi€n cuu vA ihu tluEe
r:hiir:g k6t q*& pholr! 1:tiir hrn n&a. Hoi': cd
tirE rirn thr3r:: rihiing quau hQ trong tam gi*.c
vi thrpe hidn nguydn t6c dl n6u tri:n dii ph6t
hiQn th€rn nhftng quan irs hoi . Lh&.ng'hgn ed
th6 n€u [Cr b{t ding thric sali riirl':
do.ltu -f drftir i- d*I?. F 2(ddc * d.d, * d"d6)'
Cic b4n hfly chirng minh tr6t ti&ng thr?c tr6n
sau. d6 6p drlng nguy6n t&c d[ n€u 116 .tirn ra
rnst tr6t d&ug tht'c m4'i.
1{6t c&u h&i duqe ilSt ra: N6u ta thr.ic hi$n
phep bi$n dbi / tren rndt bdt dfi.ug thr'rc n&o dir
li0n ti€p h-ai ho[e nhiEu lin ihi ti6i1 ehirlg ta
r:6 thu atg-c ltdt qu& gi mni nira kh0"ng ? Criu
tr& ldi iI: lihorrJ. I'hEe hi€n phdp bien tli5i /
ti0n ii{p hai lS.rr thi ta sB tr6' vE b{t d$,ng
thuc han tI.\u" iirlLr]'llu irh{n cira di6ir ir'r iiuqr:
tim 1lr.{1: & tinh ririt ddi trgp c{ra 1,hep nghich
rlao; n6i rii horr 1*: u;iu la thgc'hi.an iitu ti6p
hri iln mQt plrep nghii:li d&s ni* dil reis rnSl
hrnh thi ta iq.i tr& vC hi;rh d6.
C.tdi cirng xin n6rt, mot gt.r'i i khAc. Ctrrg ihr.rc
hierL pbdn nghich d&o pqc ilf , phuo'ng tich 1'
<li& su' qua phep nghich dEq d$ c:i6 hinh chi{tr
r11, 8t, C1 cfia diBm -rll tr6n c&e c+nh 8C, C"{,
l.E ttreo ttru tr3 biiin thinh c6c di8rn .{i'' .6i,
rii. Ila;r' gi& tai x6l liim giir: :{i Ai Ci eiing
vdi diEm rid. 'fa lri;l tim quan hQ gifi'a c5*
khoin-g.r6:ch iii ilf d6n c6"c ilir:h vi c*c i:4nh
trong hai tuni giAc 'tBC vn .Ar'Br'Ct'. Ta ili al6n
nr$t phdp hidir rl&i ni6'i vq cho * t{c df l5 r vAo
c'ir'r: bat dSng lhrte de bidt. \'ii:e tliirr: hi$u ctt
thF ,cin rlinh cha lrq n di.rc.
'+\
N${tfNG BAI
tiA,,cic dE thi vio Uqi hqc kh6i.A vd trong
tuy8n t$p (( Nhirilg bir toirn so cdp, f*p 3,
Hinh hoc ') (\iri rrr{t bhn t}tl vi ', lt(,N,
1-976), c{c i:g.-n riii gip in0t siJ bli todn eo ii€n
quan Sdil rn{t *iiu. Chrlng toi rnu6n girii thi'$u
lh&nr nr$t s6 trti khiri: dr cdc Lr&rt lu-r'0n tlp, nltSt
ln d6i vrli l{e .bal r} xa'lhri 116, nhi}ng noi mir
tii liAu vE to:in kh,3ng cr1 phiBrr
1. fn:#c tiirr, cln phiu nflm vfrng sr) iliEu kign
xac ftiuh r*Qt u:{t ch.u le 4, *hfr.ng h4"n cdn phii
edc ,:tir:ir t$.rn i.l.i ,lilu kigr:i vir bdn kinh (1 diiu
k:On). tisu hi$t rrrtrt c*.u r'ii qua rrst di$m ii'c lu
bi6t dusc t diBu krQn (c,\t- /i); ndu biet mf;.t
cln t.i{p xirc v&i in,;i rn{t ph8ng trrc lir bidt 1
VE MAT E&U
TnAi 'i'i{!i{ 'iP.iNH
ili6u kiQn (d:-R); alu bi6l m{t cb.u ti€p xtrc
viri mgt rriSt clu tric bi6t duEc ! iliEu ki$* (d:
/i + iR')... Ti.,n cia mEtcflu it! qua C Cilnr A, B,C, D
hlidng dbng phfirig c6 th& x{e din}r ltru'sau: l6y
gieo tliErn cfia.trge tri& lam gi6c dBC (tuc li,
ilud'ng vuOng g6c vdi m!.t ph&ng ;lEC v& di qui:
I g.m vdng trO$ ugo4i ti6'p t arn gi&e d8{-')
va m{t ph&ng ilung tr1:a cia :{Lr (hoec c'ir
BD,,CD,
2, Sau ,I8.y, ,l-rting tdi gi6r liir,-1u cung cde bgn
rn$t s6 bii tf,p bb tch vb mXt cHu. (.ir b+n hE.v
bit tay vio nghi6n e uu vrQc giii quydt d3 lr,r'
tQp. Chi khi nao khQns giiii duqc, hi ci
rrhrirns'ch? d6n 6 phln 3. d.it,
TAP
l#
1:
s-
lt
r'-r.: "*il"
\'., - r+r,
\ .-
SAi !.. Ihlng rn{l chu tli qua 'J di8nr A' }f'C
tU .io ""fl- ,+i *e t rri{l.eH*-{{-i) cho rufrc; -
tsAi 2' M$t rnlt b[u hin kitir r ti6p'xfic voi
-a" ;a\lrra" *au t-r"r, ch6p lam gi{c tei tty: ta}]
;;;; *ut ev. rbng 3 g*c 6"dinh cill :ti"T4t
;;;;r-; " riel' tinir cantr u6n crla hinh'eh6p'
, tsa! 3. Di6n tieh xuug quanh ":: ,*?-t lll^1
chOp tam $i{e hfrng S, ehu vt 66y }ing 3c' }t$r'
,i+t'.i"-it[p xric v'*t- 3 t"4nlr c!a.tt"ar tqi rung
ai6,r, *6i cauh vil cit efc cgnh bdn ts"i troog
iie* *At cinh. ch&n$ minh. r5ng hinh cbdp ld
ae" ve ttml'b6n Uinh.m[t s[u' :
Bsi 4. Cho hinh btrop tu giaee'encn vil hinh
*a" "Oi-tre;. il, ABdD ta uinn th*ng can (dB
ii ,,*4, nti6) co canh h€n bllg I vir g6c nhoa
;;;i,.'ru.J *+, ii€n IPD vi rlpc l! .!at tam
;i# ";t.l
( Atr' = PD; BP = PCr*tao .'.6'" d6{ :l-'i
i6t gO. ct. "Ilnh b6n kinh r tta hinh eiiu r81
ti6 p.
XUi 5. Mqt htnh ehu (S) tern O ti6p -xrlc vdi
mat pU&nc (Pl tai f' Tu rnOt iiiEnl. bf,t.kj. M tren
;;, ;ffhei t,iP tuYdn eit (P) t+i A vi B'
'a) So e0nh hei g6e Ail[B vd AIB-
b) csi I' le diEm &di'xriug ciiri r qua d8'
i.-:hunp mich r&ng #;;ii"; A,'{' 1" M "'i'*tg n[m
,t"ou'*0, met ptr&ng vil IIJ' luOn lu6n Ci qua
*oi-u,E*'"0 glrit, ttt''i rr rli chuvBn tren (S)'
SSi 5. Cho i{u&ng trdn (C) iA'm O'.d.u6ng kinb
'an::zn. eqi (S) la'Ltlt chu nhan (cl tlm'dudlg
i.-ur, ls;t-("sl ti6p xrlc v&i m4! p:i"c-(P).tai a;
;";;-tt,hrg (d,'nnrn trong (P)' c{eh I rr0t tloan
bhngl*o':I'L
e) Chring minh ring ti' s6 -tt/r bd nhdt (:3)
khi (7') lil"d6u, trgng iio fi i f{n,luqt ti bin
hinh m{t. c},u n$i ti6p vi rtgo4i ti6p.
Hly thu tiep bnl-toAu tr0n vE m{t ph&ng
BAi .8. G*c tam di$nr dinh D trong ttt diQn'
,,IBCD lil vn'Ong. Gqi O IA thm m[t chu ngoqi
ti6p tu &i+, ay vd l1 li b6n kinh eria u6'
'ir) Chrlrg minh-riug tbn lrri m6t tam giie mi l
b) Gqi S lA. diQn tleh cria tam $i6c gy'D,1^ n: i-
nn-d DC= c. Chitng miuh ring aDc = 4liS'
BAi 9, L6y hai diph d6i nhau trong 'htnh lap
uhuons. rbi, n6i e{c trung di6m cfia c{c canh
inono?i huu hai ttirrtr iiy, tqo thlnh mSt hinh
tge gia. d€u : thi6t di$n nAy chia hinh i{p phr:ung
ra ilm hai phhn. Tim l:6n hinh m5t chu nQi ii6p I
trong m6i pira.. tlr"o cqnh tt c&a hlnh lSp phuong'
Bei f,O. ba mdt ehx c6 chuug 6ibm '{' Bi6t
rilng kh6ng ed dudng thEng nio qua { la^i ti€p srlc
vdi-cA I m4t cau 6y-, hEy chung mintr rtng 3 miLt
clu di oho'cda c6 rn$t dibm chung thu hai nira'
t.Uac;.
Bii,l-1. 'tt m6i 6inh c&a-mQt'tir diQn d\rng
m{t ph&ng chua t6m tlucrng tldl ngory ti6p eia '
*at ifOl d-i€n vd vuQug gdc vdi m4r 6y' Chtlug
'minn riog 4 m[t ph8-og thu trdn giao nhau tqi.
t$m rn{t eh.u ngoli ti6P tfr rli$u:
BAi L2. Trong tanr'giot vu6ng, ta c(s r:p-rt'-
v6i r ta bin.- kinb tlqong tr0n nQi ti6p' o la !
tuong tg trcng:khbng gian.
"a) Gei L{ lt mOldiqm $1"}-1-'ET tran (d}' riru
qot ni;n giao di6ni M,' crld '*.": *U.t :u"._,
I.,r) Quf tirb c'l a il1' li' nrQt dudng tit)l' Hiy
tiJ; #; kinh i cfra no rir tirn tfirn I cila dudng
trr.ln ii )'.
I c) Tinr'qrii tieir tira t ktri (cl) tli'ctrirlFn sao
. "lr;'";'"A; t'i$'p x*c't1*i-rlrtcrng frdl tam fJ ]:6n
kinh c.
bal z."cpi.o ie iam rlrat eiru n{oai ti6p tu
rlign {?'},: '' d'ecn: G, "Gr, ti5, G"' Ga l}rr luq! li
;;;s';d,r', c,ra (r) v'ir cira "'-*ttYlat d';i ;irfn.-riintr
' ,f,'g,i,13, ro }ir t*tn qr{t c*3 di qua G;GuGrGa;'
;;;il; chi6u vuQ-ns P,oc cria o't:'' I? l+n m{t
pl;-rr! ,tse I&n luqi ia ua, Gl' r':s' Ila; giiro 6!3trr'
I *,1*-iil*,-.&i ni6t e$u ii -q; -+
n). Chrlng rnierh ring u)o'd : t lt!llt)"
' b) Ctrung minh rhng ndu {r''r) iidp. xqc v&i mQt
*4,'"I; ji: ,rrr ar,'t iil* t" c6ch dlu b tlou "'ia
:r tir.-:o xric. l,l6u i*)ti6p xirc l'Si:2 mXt cria (?)
'e'"rg tT) c6 5 cqnh hfng nhau;-ndu (o) ti6p
i ui+t thi (1r) ii tu diqn dEu'
. BAi 13. u) Troug ,tam giilc '{-BC' chrtag minb'
ring ndu tflrn &udng trdn hQi ti6f tlnng v6i tAm
auing trdn ngoai tiAp.thl AB'- RC = C:l va trcng
tAm cIa tatr giic ciirig trring voi hai tim noi tr0n'
h) }[iv l0ag eho lrud::g hq'p tu cliQn: r"6ri tim'
ii4t i'Lu tOi-tiep lrirng--v0i li*."1'rr{t cdu
ngogi"iiJp thi c6c [n6t cria tu iliin blng nhau va
tl!ng ta'n cia tr! tli6n trirng v6i.hai't&rn n6i tr'0u'
e) ,Dune tit tli"l* ntr4n nr{t clu (si vdi bti*
!irirh r lirn mEt chtr uQi ti6p vA nh{'rr. m{t +hu
iS) vsr birn liilh It I&m rnlLt chu ngoqi ti6p' trang
tlo l? ) :ir.
Ent 34.-r:nq'tu diEn A1:4,z.43Aa Chtng minh
rd *g e6c cgnli' f.i,44, As 'te; vI ti trerng thlng ndi
i-r,i..,iig;lflrn G voi i6r* rl$t ehu nfi tiip e ring soug
.onj'rui' rnQt rn{t Pd&rig' '
3. Ilh&ng ,{ea gi'&i ragt'sd &'*i tr:{q ban
irlY t1s vE hinh)'
Bai 1, ]-am .t nfr.m tren lrl.r7 (d) ctra.tam giric
XAC,'trcp 1]ibm D:ii1m trern O1' U4t.ph&ng (O,tl)
+
'i,i
u
ii tritt pli&ag tlsi .";iing. r:&i nr5t cdu tlf; r:1:r.i theo
' dir<rr:g trdn 1#n r{-) vi cii,l mlt cSu cSn dr:.ag
tlreo du<'rng tro* lirlr (n). G+i riil, ]' ]a giao di6rn
giila rndt ph8.i:,i (U, r,ri v&i :iuctri; rrr)n di qua
rl.BC; !:ai terir ir& vE d:rng: riurg dur"iilu lrriei
{D) rli qua r"i.;\' vi ti6i: l*c vci i{l).
H;&r *,.{;.}., ti ,. fI::, i{r lJ: lru.t" ll,,r ri;r r"*Et ll.,i.JJ,
t)iit,, |j,1.{;; r-}Ei ,i_..,x.ts, flfl3-j_;ic. Erii:i_/j,l; f}1,'* I
lxl. i, hri r: r i n B .Ei/ l : &ff 2 :ll il a, &,rit -.t'l I { 2: B i'i t,.
]iFl c{c iarn sii.r Liiug niir rr i}-4t1:, DEtlr"; I)BEi,
1)il.FJ1; DIli:1, i]3i:r; $S,1. fi8il ,J$ *rr dIi-.=8{.', },1'*-;
"/t' /)C. 'fuer:g lr::,8{l:il.1" l}.5 :l-1 ,i. t1:* tam giic AflC
:V, li il6ir y;.r i.i-,{,::fl,r}:i;il . i-.;-iri2--:]J-iti. T'?rsr {J c,ia
E' l:liil {r;iu irci: iidp iiirr, ii'ir: iinirng cao 1l-f,j'. II*i
tanr $i*e {-lJir,;-j" l'.':il'D iib-rr; ii4ng rtoo {rrrr/i,lrfr" ,,.
IIzDlf)'D T:{1i't I,:}{)' =.r'r; ;\l;}. si* .{/fi,'D' D =* {.i 3) x
,: i/fc.rz'z A;;: tlzi) =:r-cos (-;.2r.r,"1,s r:x Ulr,il,
dung ilinh lir liam sii'sin iror.rg tau-r gi{r: DI}ti2" i;ip "
!'enib("1 :i
It : '--*t'''"':') - 1i+cosz{cl li) -:'
EAi'3. Tru&r fr .YL cNru;:g minh rang iam gi6*
dd3'AEL' ii fits,r. TlLn 1 cill ,l'*,r,,ug tron ndi iidlr
vi l{u'&ng trr..r: rrgo:,ri t i6p trririlq rrhau. Gqi A' ,8,'{.:'
i& d rlrn giun riiii r:,in?r' l],i, llli,+t,{.'. $rrd':rg trrin
ryua .l'-B'ii' t$ irh c1d.r,-iir#i'!i?, ir*r.l.riJ ti.ong
phC,,, r'i i.,r lr'ri;r -lj, ri r;5 \i2, n$l r-:ai irriri
ph&rig (.rl'8'C") r-n (,1$t) so!tg sr)ng. '{'Arn ,tr' cril
11rrr.:ng irdn .d'6'{', I .vi l} liring hinq. 'i'Am nn5.i.
r:}ir n{m il'.1r"r l-{' i'u,iu5 g(rc riii I'rai mdl ph&ng
noi tr&n. llf,-.r I),,18i) lh hlnh cliop tarrr gif,c dEu.
Gr:i E la rliilrn tr6p rtc tr*n CB; xdt thi6t di6u
S,{D dE tinh r.
iffi-il7
B*i 4. &ai L vn K 1,i"
':4 t:
ir*.:f.g ii:iim ri:r ilC lrr
s* /c. \"fl5r q:r! ii,llr e*a l ld ilurlng tr.&ir inh,:r,a
itf tiong ph6p vi tu t&rn A ti- s.i 1{.
BAi ?" "l'rong 1gr11 /J cilr f r? di6u cdch r{iiih
- bAu{ 3/4 irung '{r.tl'6i: turyrg I'r*51- 'Iri" ci*, ta ctr
e@ € tri+ l!,,,*$! sr$4>
fi ioirl.l + AO$:::{{jg'4 =-- D"F.ra: l.tl;l) _Ii,X,-r f;Og"
IS *,.. : ( j16) i-rii'. i'1:u hrp troii5 nril ;:tr8nq :
olr:)6 = {l/4),1,{' {.cjri6Lr jir* f;i,}.
Bni S. tloi H, FI" I- iiin lirq;.i iri i:h:i* ,::rl
;iLl'dr:1 *ur.' h4 i&' J). C, Ii r*dn;q duciig kinh,lF'"'
, ., L. h-.:
t-ath i; iie ,8,5 soiig :ion:1 r,a bin$ **. 11.';ir,:i Li
eiia i:i5.t t.iri, ng*1i tiep t* aiea it dibrl giire
r:ie lB'. iii:,1.,: scng o{rr'g r'f, h*rLg .D{-:, nEri
it;L' BB' l.t irinh :;tr:h hlntr. i{6 li},r-l_j8", v*y
.(.\:l:,1]. lid chir:rgi rninir litlp D.N = CLI , iric tiln
tgi iarn 8i5r {rjf.\r '.hiiu rrEn,riIir L;.iri {r.: I L"ri
iirm Bi:';* Clfl'vd rS'Alt Xr{;i0:l'E),. {i/3iS"2,$;
i"r i'lJ..l3t : (r,iri) ul;::. Ilr'-rng i: {,\.jJ,l'jj} :
:1- (A.llA'B), vir rbr; .= 4,115,
' Cach 2. *ei .llK = l, CH = nl, -8,L: n; il'$*t{
.ram gld.c ilAB' eb I t . | ;2 + b4 +;' * ,rri i,'-+l?
q:io 116 t * ,f *:;;?- I '/7-+ b\:;?
t,
tn = bi cz * u' I ./;" ;*{rz + rr, i.r : c /;i :;-;,!
./-.f---=--== -
1r" r l,! a cz. ila th{-v r&r:g (m -- ,,)2 .1 l? <
(rt , n'," '.:r2 ,,1,, .-:,; .a 1 !tn I'1,', l: ,: .! l.l .rjitr
[1Cl: r:i.t:r liirii {ijr* I]i."-', rl,i t{]5? == il -i ln t l)X
(- 1 * iti -r nj ri - rn -i *,) (/ *;n -- f i; f 'U'r'I
eaz + irz *"?i: ,tth212tR2 lue olc = {r?S.
E*i LE. 'ilr'truc ti+n, chLing nrinh r5nq l,nrrg
ha m{t c$u dfi. cho kirfing c+ 2 rfl+i eEu ii6or
x*e v&i nhau. ?hiit "rgy, gi5 sr? ed 2 m{t e&u
li$p sir* r&i rhau tqi C thi ti6p diBm chrlng
qua .,{ {Al !:*[i chu iher: m$t ;]uong tro*, 'f i{yi'
trivd* l+i d cti* dLtd'rg tri)r 6y tidr trir: vgi rL
.i m{t c[u, tr{l vdi giA thi$t. Y+y ? tral"l 3 rnEl
r:3'Lr pi:iii e61 nhau ihr,o m$t drr#n6 tr'O& €i ({ilr,
.i. ilu*rg tldn nriry kh6ug tir& ehi c* mgt diin:
.t rhu*g vrli n:ht c&u thu 3,. ti n$tr r'[;.' i[:i
ti$p luydn tai ;l efr.a.duo,n5g ilSn fli ti+ip xrle v$i
cl l:a rnEt ciiu, trl1i r6'i gii !hif,i. {iir.le"r.r.iiEr*
(kh6c ;1) giira dmcrng $ron 6y r.6i m{'t ciir: tl',{t
3 n&rn ir$n ,3 n:f,t chu r1{ cho.
Bei tE, L)!a:;.= --Lls:1,{i-:$
a -i'b -i ,' ' ir'"g
{Id 5 i.a disn t"ich rn*i i:.i:yEnl Si (;:1. 2, :l) ia
tiigu tii:ir cfla ;ndi btrl ; n, &, c tS ha *q::h cr':*
goe tilnl d ien vudng.
. $A;, 1:!" h) Tr\ gi,1' lli6t su-!, j.:, r',tr,- r{.,
dn'}eg trc;: ng*tri ti6p c{c m4t d&u }:*ug nirati.
Cu tung c[p t du&ug tron iai co CS; cu' g *hring
nsn i!'6rrs t? ry .*"j:" fA,rlu-,1.".9.d1-iba:l{
ghau. t1*i -,34C : .$O{l .=1, ,i8g '= ,4..{-}g:= -
Dso: .r'= I
'!'
ll,I"'tr'rrtr:s irdt ctr,'r;ig r:riirl.:-ring J.Pl( la gd;r
phing *ri* r:i:.i _ciii* (i?EC, il.L,,). TleDi gir:c PLi{.
c$n, nda du*lig r:ao Pii etilq ii dr:drr:g thdi;r gide,
nr l:.fry Pf:I n&r."r irong m'at ph$ng pii&r gi*cr eta gSe
: nt i OiCn i:i,l trin, 'i'idp dti *lrri.i:g t*inh r$*g p,{
& cfiug niir* ire*g rne[ ph$ng phin gi;ic rira ge]c
nh! di6n if,4g. pcll) bing cietr chrii;g r:rinh i&cg
h$t i(-i; Si&m ,ii l:.*: n*m !r*r_:g ;nf t nlr8;tg [,"D.ltl
cttirg *l:.ch iirim hai rrii ph6*g (F;,ir i, iFCII) T'ir dS
dA tin,rt r" 'Edp : ;:: I(f/ tg (a.l)=(;lt) *iuif tg (e,/3).
BAi 6. Ci:u y.r,rJ'. ^i]1 :,1,t2. Iihi ci l t.uy6u
. A,hI' kyr: nh$i tiri bi6n thirnh drr#ng tri,:rh. AiU'
kyn irhlt l,hr Ali l-15n rill€it, iii,* k!ri B,lI ns6.n
nh{i (.}? trr\ng l_:i;r e,hA:r ciudu6 rrri.n! gOo hg trl.
B xui5ng trl)); Aiv[' * qnztt/+A{+n2*, r x 2Rz!
ilF;{Ta iu6n e6 AilAkt = 2ga1/.igy;*,*L.
tue / ls Snir cria Jlf, trong phep ui {u thrrr,{" tj-